Autocatalytic rate equation

Equation (2.11) shows a common autocatalytic rate equation proposed by Sourour and Kamal (1976)... 

The chemistry for a stoichiometrically balanced reaction suggests that m = 1 and n = 2 in Eq. (2.86). For real systems, values are often close to these values but not identical. In the epoxy-amine reaction the alcohol, which may be present initially in small concentrations but is also a product of the reaction, catalyzes further reaction, resulting in autocatalysis. Since there are four unknowns ki, k2, m, and n) nonlinear regression analysis must be employed, although ki can be evaluated independently as the extrapolated reaction rate at a = 0. Autocatalytic kinetics are usually evaluated by the derivative form of the autocatalytic rate equation [Eq. (2.86)] with data coUected by isothermal method 1 measurements. Activation energy E and preexponential factor A are measured from the Arrhenius equation... 

The first kinetic study of pepsinogen activation was made by Herriott (6), who showed that pepsinogen is capable of spontaneous activation if it is subjected to an acidic pH of 6 or less. At pH 4.6, 25, an 8.75 mg/ml pepsinogen solution is half activated in about 6 hours. The observed data for pepsin concentration as a function of time fit an autocatalytic rate equation of the form ... 

This is an autocatalytic reaction, in which a product of the reaction appears in the rate equation for the forward reaction. In this case the mass balance expressions are... 

Initially, A = 0.5 mols, B0 = 0 and D0 = 0.05. The reaction is run at constant pressure and temperature. Given the data of the first two columns beteen -the rate and the fractional conversion, confirm that the assumed rate equation is correct. Also check if the plot of rate against concentration has the peak that is characteristic of many autocatalytic reactions. 

An autocatalytic reaction may be able to proceed in. the absence of the catalyst. In some cases the catalytic product may be removed as it forms, by distillation, extraction, precipitation, or some other means. The process, A => B + P, may have the rate equation... 

Because of the particular form of the rate equation (like that of an autocatalytic reaction) no reaction is possible in batch or plug flow when Cx0 = 0. Inoculation with product cells must be done at the start. Practically, however, the process may get started by itself after an induction period, since the theory is not exact. Values of x for several values of Cx0 are tabulated. 

Steps (I) and (II) are autocatalytic. The concentration of is held constant by supplying it to the reaction. This leaves [X and [7] the concentrations of the intermediates, as variables. In this case we can solve the rate equations exactly for the variable concentrations of X and 7, but concentration of A is held at a... 

Malkin s autocatalytic model is an extension of the first-order reaction to account for the rapid rise in reaction rate with conversion. Equation 1.3 does not obey any mechanistic model because it was derived by an empirical approach of fitting the calorimetric data to the rate equation such that the deviations between the experimental data and the predicted data are minimized. The model, however, both gives a good fit to the experimental data and yields a single pre-exponential factor (also called the front factor [64]), k, activation energy, U, and autocatalytic term, b. The value of the front factor k allows a comparison of the efficiency of various initiators in the initial polymerization of caprolactam [62]. On the other hand, the value of the autocatalytic term, b, describes the intensity of the self-acceleration effect during chain growth [62]. 

When the rate equation is complex, the values predicted by the two models are not necessarily limiting. Complexities can arise from multiple reactions, variation of density or pressure or temperature, incomplete mixing of feed streams, minimax rate behavior as in autocatalytic processes, and possibly other behaviors. Sensitivity of the reaction to the mixing pattern can be established in such cases, but the nature of the conversion limits will not be ascertained. Some other, possibly more realistic models will have to be devised to represent the reaction behavior. The literature has many examples of models but not really any correlations (Naumann and Buffham, 1983 Wen and Fan Westerterp et al., 1984). 

A systematic study of the validity of such a procedure was performed in collaboration with ETH-Ziirich [15], The validation of the procedure was based on numerical simulations of dynamic experiments and adiabatic runaway curves. These simulations were carried out using different rate equations nth-order, consecutive, branched, and autocatalytic reactions. Moreover, the results were compared to experimental results obtained with over 180 samples of single technical chemical compounds, reactions masses, and distillation residues [17] (Figure 11.8). Thus, they are representative for industrial applications. The line corresponding to this rule (Equation 11.5) is also represented (full line) in Figure 11.8. All experimental points lie above the line and the safety margin remains reasonable. Thus, the method is conservative, but delivers a reasonable safety margin. 

This procedure for estimating TD24 from the temperature at which the peak onset is detected in a dynamic experiment is justified, since the TM Rad is based on a zero-order approximation and at the beginning of the DSC peak, the conversion is close to zero. Thus, the heat release rate determined by the procedure is not affected by the rate equation, at least for non-autocatalytic reactions, and may be used for the estimation. 

Abstract Theoretical models and rate equations relevant to the Soai reaction are reviewed. It is found that in production of chiral molecules from an achiral substrate autocatalytic processes can induce either enantiomeric excess (ee) amplification or chiral symmetry breaking. The former means that the final ee value is larger than the initial value but is dependent upon it, whereas the latter means the selection of a unique value of the final ee, independent of the initial value. The ee amplification takes place in an irreversible reaction such that all the substrate molecules are converted to chiral products and the reaction comes to a halt. Chiral symmetry breaking is possible when recycling processes are incorporated. Reactions become reversible and the system relaxes slowly to a unique final state. The difference between the two behaviors is apparent in the flow diagram in the phase space of chiral molecule concentrations. The ee amplification takes place when the flow terminates on a line of fixed points (or a fixed line), whereas symmetry breaking corresponds to the dissolution of the fixed line accompanied by the appearance of fixed points. The relevance of the Soai reaction to the homochirality in life is also discussed. 

As the first step in analyzing flow trajectories in a phase space, we consider the simplest case when reactions such as spontaneous production (Eq. 1), linearly autocatalytic (Eq. 2), or quadratically autocatalytic (Eq. 4) reactions are active. Then the rate equations are simplified as ... 

These observations can be represented as a special case of the general rate equation derived by the application of order-disorder theory to diffusionless transitions in solids.3 According to this equation, the shape of the rate curve is determined by the relative numerical values of zkp/kn and of c. The larger the factor is relative to c, the more sigmoidal the curves become. This is understandable since the propagation effect which is responsible for the autocatalytic character of the transformation becomes more noticeable when kPlkn is large and c small. Under these conditions some time elapses before a sufficient number of nucleation sites are formed then the... 

Apart from the linear autocatalytic ansatz, the rate equation involves some further assumptions the relevance of which should be discussed. First, the neglect of production terms other than those due to template-instructed reproduction seems straightforward. Whenever template instruction can become effective, it soon will outgrow any other type of production as far as the formation of specific sequences is concerned. Second, the assumption of a buffered level of substrates may seem unnatural. We have studied the effect of exhaustion of substrates by the upgrowth of a more efficiently reproducing mutant in a medium for which the influx of substrate is kept constant. The... 

In the mid 1970s, Falconer and Madix observed a surface- kinetic explosion for the decomposition of formic acid (HCOOH) [23] and acetic acid (CH COOH) [24] on the Ni(llO) surface, characterized by very narrow product desorption peaks in TPRS. Such autocatalytic reactions have also been observed in the decomposition of acetic acid on Pd(llO), Rh(llO), Rh(lll), and even supported Rh catalyst by Bowker et al. [70-75]. In general, these reactions exhibit accelerations in rate as the reaction proceeds to completion. Earlier work hypothesized that decomposition of the carboxylate species formed following adsorption of the acids on the surface was initiated at vacancies (i.e. bare metal sites) and propagated by the further creation of vacancies as the products desorbed from the surface [23, 24]. The rate of decomposition was well described by the rate equation r = -k(C / Cj )(Cj - c+/Cj), in which C is the instantaneous surface concentration of carboxylate, C, is the initial surface concentration, and/is the density of initiation sites. Since the decomposition produced an ever-increasing concentration of vacant sites, a kinetic explosion occurred. 

Furthermore, it has recently been found that the discrete nature of a molecule population leads to qualitatively different behavior than in the continuum case in a simple autocatalytic reaction network [29]. In a simple autocatalytic reaction system with a small number of molecules, a novel steady state is found when the number of molecules is small, which is not described by a continuum rate equation of chemical concentrations. This novel state is first found by stochastic particle simulations. The mechanism is now understood in terms of fluctuation and discreteness in molecular numbers. Indeed, some state with extinction of specific molecule species shows a qualitatively different behavior from that with very low concentration of the molecule. This difference leads to a transition to a novel state, referred to as discreteness-induced transition. This phase transition appears by decreasing the system size or flow to the system, and it is analyzed from the stochastic process, where a single-molecule switch changes the distributions of molecules drastically. 

An alternative mechanistic representation [13] of autocatalytic behaviour [5,8,14] which results in a rate equation formally similar to the chain branching model, assiunes that stresses produced by reaction cause cracking of the reactant crystal that increases the surface available for nucleation. These equations are of the form ... 

Only the first step is autocatalytic as indicated in the first rate expression. If the three rate equations are added... 

The kinetic experiment does not distinguish the mechanism between (9.17) and (9.18). The kinetic expression also indicates that the PPy grows via an autocatalytic niechanism as was the case for the PAn, although the mechanism is not apparently shown in the growth cyclic voltammograms as in PAn growth [42,46]. Finally, the polymer growth should ideally takes place in one dimension as the rate equation has about j order dependency with respect to the amount of polymer. 

The hydrogenation kinetics of cydohexene catalyzed by Pt2(dba)3 dispersed in BMI.PFg, BMI.BF4 and BMl,OTf are shown in Fig. 6.3. The kinetics curves were treated using the pseudo-elementary step and fitted (Eq. (6.1) by the following integrated rate equation for metal-salt decomposition (A —> B, hi) and autocatalytic nanoduster surface growth (A + B — 2B, 2). For a more detailed description of the use of the pseudo-elementary step for the treatment of hydrogenation kinetic data and derivation of the kinetic equations see elsewhere [81-83]. 

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